Cartesian System of Coordinates

Answer

Step 1: Use distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ Step 2: Substitute values: $\sqrt{(7 - 3)^2 + (1 - 4)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ Answer: 5 unit…

Answer

Step 1: Find AB, BC, and AC. AB = $\sqrt{(4-1)^2 + (5-1)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ BC = $\sqrt{(6-4)^2 + (7-5)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$ AC = $\sqrt{(6-1)^2 + (7-1)^2} = \…

Answer

Step 1: Use section formula (internal): $x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2}$ Step 2: $m_1 = 1, m_2 = 2, x_1 = 2, y_1 = 3, x_2 = 8, y_2 = 9$ $x = \frac{1(8)…

Answer

Step 1: Use midpoint formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Step 2: $x = \frac{-4 + 10}{2} = \frac{6}{2} = 3$ $y = \frac{6 + (-2)}{2} = \frac{4}{2} = 2$ Answer: (3, 2)…

Answer

Step 1: Use area formula: $\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ Step 2: Let $x_1 = 1, y_1 = 2; x_2 = 4, y_2 = 6; x_3 = 7, y_3 = 3$ Area = $\frac{1}{2} |1(6 …

Answer

Step 1: Apply area formula: $\text{Area} = \frac{1}{2} |2(7 - 11) + 4(11 - 3) + 6(3 - 7)|$ = $\frac{1}{2} |2(-4) + 4(8) + 6(-4)| = \frac{1}{2} |-8 + 32 - 24| = \frac{1}{2} |0| = 0$ Since area is 0,…

Answer

Step 1: Use slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Step 2: $m = \frac{-7 - 5}{2 - (-3)} = \frac{-12}{5} = -\frac{12}{5}$ Answer: $-\frac{12}{5}$…

Answer

Step 1: Use area formula and set to 0: $\frac{1}{2} |1(4 - k) + 3(k - 2) + 5(2 - 4)| = 0$ Step 2: Simplify: $|4 - k + 3k - 6 + 5(-2)| = |4 - k + 3k - 6 - 10| = |2k - 12| = 0$ So, $2k - 12 = 0 \Rig…